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Find the values of k for which the following equations have real roots
$4x^2 + kx + 3 = 0$
Given:
Given quadratic equation is $4x^2 + kx + 3 = 0$.
To do:
We have to find the values of k for which the roots are real.
Solution:
Comparing the given quadratic equation with the standard form of the quadratic equation $ax^2+bx+c=0$, we get,
$a=4, b=k$ and $c=3$.
The discriminant of the standard form of the quadratic equation $ax^2+bx+c=0$ is $D=b^2-4ac$.
$D=(k)^2-4(4)(3)$
$D=k^2-48$
The given quadratic equation has real roots if $D≥0$.
Therefore,
$k^2-48≥0$
$k^2-(16\times3)≥0$
$k^2-(4\sqrt3)^2≥0$
$(k+4\sqrt3)(k-4\sqrt3)≥0$
$k≤-4\sqrt3$ and $k≥4\sqrt3$
The value of k can be represented as $(-∞, -4\sqrt3] U [4\sqrt3, ∞)$.
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